Space coded linear array antenna



Aprll 21, 1964 F. s. GUTLEBER SPACE conEn LINEAR ARRAY ANTENNA Filedoct. 23, 1961 l2 Sheets-Sheet l NNK April 21, 1964 F. s. GUTLEBER3,130,410

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SPACE coDED LINEAR ARRAY ANTENNA Filed Oct. 23, 1961 l2 Sheets-Sheet 6FRANK s. Gun een ATTORNEY April 21, 1964 F. s. GUTLEBER 3,130,410

sRAcE coDED LINEAR ARRAY ANTENNA Filed oct. 23, 1961 12 Sheng-sheet 7INVENTOR. FRA/VA S. G 'IJTLEE/Q BWM `/4 ATTORNEY April 21, 1964 F. s.GUTLEBER SPACE conED LINEAR ARRAY ANTENNA Filed ocuzs, 1961 A l2Sheets-Sheet 8 Q .S Y E -m l I I o f a I`- w w. f

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ATTORNEY April 2l, 1964 F. s. GUTLEBER 3,130,410

SPACE CODED LINEAR ARRAY ANTENNA Filed Oct. 23, 1961 l2 Sheets-Sheet 10A 0 s 2 nv k *C Q u 2, 9 w

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coDED LINEAR ARRAY ANTENNA Filed oct. 2s, 1961 12 sheets-sheet 11INVENTOR. FRA/VK 6. GA/TLEHER ATTORNEY April 21 1964 F. s. GUTLEBER3,130,410

Y s PAcE coman LINEAR ARRAYANTENNA Filed OCT.. 25, 1961 l2 Sheets-Sheet12 FRANK s. gant-afk ATMRNEY United States Patent Oil ice f 3,130,410Patented Apr. 2l, 1964 3,130,410 SPACE CODED LINEAR ARRAY ANTENNA FrankS. Gutleber, Wayne Township, Passaic County,

This invention relates to antenna systems and more particularly tomethods and equipment for obtaining any desired antenna pattern from alinear antenna array.

Array antennas of a variety of types are well known in the prior art.These prior art antenna arrays have been of several types. One type hasutilized a number of equally spaced antenna elements to build up anarray. This type of simple array of equally spaced antenna elementswhich are provided with equal amplitudes of driving current or voltageproduce an antenna pattern whose shape can be controlled to only a Verylimited degree. A typical response of such an equally spaced antennasystem with equal amplitude driving power at each individual elementresults in the well known sin x/ x radiation pattern. Although such apattern can produce a narrow central lobe, it suffers from the defectthat the irst side lobe is quite high in amplitude compared to the mainlobe, and in fact the rst side lobe is down by only 1,4.72 orapproximately 13 db. The values of the central or main lobe and the sidelobes relative to each other can not be changed no matter how large anumber of elements is provided in such an array. The relative values ofthe side lobes and the central lobe are xed by the geometry of thesystem. To' improve over such equally spaced linear array antennas, theprior art has resorted to a number of expedents which have been onlypartly successful. Schelkunol, for example, in United States Patent No.2,286,839 shows an array antenna utilizing a number of equally spacedindividual antenna elements. Schelkunofl` provides both an individualamplitude control device and an individual phase control device for eachof the individual antenna elements that compose his array. In generalthe amplitude of the driving current which is applied to each of theantenna elements is different from the amplitude in other antennaelements. Likewise the phase of the signal applied to each of theindividual antenna elements is in general different from that of otherelements in the array. Schelkunoi'f provides a systematic manner ofchoosing the particular amplitudes and phases supplied to the elementsof his array. However, these amplitudes and phases follow a polynomialdistribution. Although this procedure resulted in a somewhat improvedantenna pattern, it has a number of extremely serious drawbacks. Themost important of these is the fact that, since diierent amplitudes ofsignal must be supplied to different individual elements, eithertransmitting stages of different individual design must be supplied, orelse large amounts of available power must be dissipated in powerdividing networks etc. to provide the required ratio of feed powers.Also, individual phase control must be supplied to each element. Inaddition to these practical disadvantages, serious vcomputationsproblems arise in attempting to design such an antenna array whichutilizes individual elements unequally supplied with power at dif ferentphases, because the eiect of an individual antenna element on the arrayas a whole is by no means readily ascertained from the equationsrequired for the design.

Other prior art systems have attempted to provide antenna arrays whichlikewise utilize the method of supplying unequal amplitude and phasesignals to the individual antenna elements. For example, binomialcoeflicients have been utilized to provide the amplitudes supplied toindividual elements in the array. Such array antenna sysd tems likewisesufrer from thepractical defects of supply mg unequal power and phaseto. the individual elements and from the computational diiiicultiesinvolved which make diilicult or impossible the actual detailed designof a particular desired antenna pattern.

A further disadvantage of the prior art systems has resulted from thefact that many of these arrays are built up out of elements spaced inthree dimensions from each other. That is,elements are placed at certainpoints within a given area and also above or below the plane of thisarea at certain elevations. The difficulty here arises when it isattempted to improve the radiation pattern -of the antenna by utilizinga large number of individual elements. The addition of new elements doesnot produce any effect which is readily predictable from the type ofmathematical analysis provided in the prior art airay systems, and infact it is often ditlicult to determine where, or in what part of thearray, additional elements should be provided.

Therefore, it is an object of my invention to provide a systematicmethod for increasing the number of elements in an antenna array tosystematically improve the resulting radiation pattern obtainable.

It is another object of my invention to provide a space coded lineararray antenna which utilizes individual antenna elements which are fedwith equal amounts of power and with equal amounts of driving currentwhich may be supplied at the same phase.

It is a furtherv object of my invention to provide an array antennasystem which obtains the maximum benelit possible from a given number ofindividual elements utilized to build up the array to provide an optimumradiation pattern in space utilizing this given number of elements.

It is a feature of my invention to provide an antenna array composed ofa number of individual antenna elements which isan integral power of thebase 2. A particular number of individual antenna elements are spacedunequally from each other so as to cause a predetermined effect on theantenna radiation pattern due to each set of elements which make up thearray.

It is another feature of my invention to provied` a space coded lineararray antenna in which all of the individual antenna elements may bedriven with equal amounts of current or voltage and wherein the phase ofthe signal supplied to all of the individual antenna elements may be thesame. The individual antenna elements in this array are placed atpredetermined unequal distances from each other according to asystematic set of design equations which allow the production of anarbitrary desired antenna pattern utilizing a particular predeterminednumber of antenna elements.

It is another feature of the invention to provide a space coded lineararray antenna whose design may readily be extended to provide a threedimensional radiation pattern in space with no further additionalcomputation. This method allows the particular unequal coded locationsof the individual antenna elements of a rst set of elements in the arrayto be extended to provide the pattern for the entire array. The entiretwo dimensional array may be driven by supplying equal power at the samerelative phase to allof the individual antenna elements.

The above mentioned and other features and objects of my invention willbecome more apparent by reference to the following description taken inconjunction with the accompanying drawings in which:

FIGURE l is a diagram illustrating the geometry of an array antenna, asviewed from the top, which lis useful in deriving the equationsinvolved;

FIGURE 2a is a schematic diagram of a 16 element array antennarepresenting a rst embodiment of my sin sin 00- for the antenna ofFIGURE 2a;

FIGURE 4 is a plot of the radiation intensity field Et of the antenna ofFIGURE 2a vs. the physical space angle 6';

FIGURE 5a is a schematic diagram of a two-dimensional array antennawhich utilizes the antenna design of FIGURE 2a;

FIGURE 5b is a block diagram of a portion of the circuit used with thearray antenna of FIGURE 5a;

FIGURE 6 is a diagram giving the relative spacing in correct proportionof a linear array antenna composed of 32 elements;

FIGURE 7a is a diagram of the radiation field intensity pattern Etplotted vs. the design parameter K for the antenna of FIGURE 6;

FIGURE 7b is a diagram on a smaller scale of the field intensity patternEb plotted vs. the physical space angle 0 for the array of FIGURE 6;

FIGURE 7c is a composite diagram of the field intensity of the arrays ofFIGURES 6 and 2;

VFIGURE 8 is a diagram of an array representing a fourth embodiment ofmy invention and showing another possible coded spacing of an antennautilizing a l6 element array.

FIGURE 9 is a plot of the lield radiation intensity Et produced by `theantenna of FIGURE 8 vs. the design parameter K;

FIGURE l0 is a diagram illustrating the preferred embodiment of myinvention, representing an array antenna composed of 64 elements anddesigned according to my method;

FIGURE l1 is a plot of the eld radiation intensity Et produced by theantenna of FIGURE lO vs. the design parameter K;

FIGURE Il2 is a plot of the viield radiation intensity Et of the antennaof FIGURE 10 vs. the physical space angle 6.

Referring now to FIGURE 1, the general relationship is shown between theelements of a linear array which are unequally spaced from each otherand showing the meaning of the space angle 0. 9 represents the directionof a line v drawn from the array along a particular direction which isof interest at the moment. The angle 0 represents the angle between x,the axis of the antenna, and the line v which may be moved about toexamine any particular direction which is of interest. The radiated wavefront is perpendicular to the line v as shown. For a linear arrayantenna, it is ordinarily sufficient to design for values of 0 which runonly between 0 and 90 degrees. Because of symmetry, the antenna appearsto produce the same pattern whether one stands on one side of the axis xor the other. Likewise, if the disposition and number of antennaelements is even and their distribution is symmetrical, the secondquadrant will produce a pattern which is the same as the pattern in therst quadrant and is in fact the mirror image of the pattern in the firstquadrant. YFor this reason, it is necessary only to work with values of0 between 0 and 90 degrees for linear array antennas. In the mostgeneral case, one would examine 0 for 360 of azimuth, but this is notnecessary here. The Viirst two elements of the array are shown separatedby a distance d1. The distance d1 sin 0 represents the ditference indistance that two Wave fronts starting from the two elements havetraveled along the line v located at the angle 0. The differentdistances VlO which are traveled by the wavefronts from the individualelements results in ditferences in phases of the radiation from theindividual elements at points removed from the antenna, as is well knownin the antenna theory. There are a number of ways to characterize anexpression for the total radiation due to an array, such as shown inFIGURE l, which has N elements. One representation which is particularlyuseful for our purposes is given in Equation l.

Where Et equals the intensity of the electric field radiated by theantenna, :1, e is the base of the natural logarithms, all of theamplitude coetiicients A1 through An 1 are either l or 0, and 5b isgiven by Equation 2.

In Equation 2, tI/:the phase of radiation; A equals the wavelength atthe particular operating frequency of the antenna; 0 is the physicalspace angle, as explained above; and d is an elemental uniform distancewhich represents an equivalent average uniform separation of the antennaelements.

The actual antenna elements of the array are in general unequally spacedfrom each other, but the distance d is an equivalent uniform spacingwhich I use to demonstrate the utility of my method of creating thearray. In all the equations e is the base of the natural logarithms.Alternately, the radiation from a linear array antenna may berepresented by Equation 3:

(3) Et: 10+ A1eiNn0+ A2eiNz+ Aaemyq; All lestN-w where the terms Nrepresent coded relative spacings of the individual antenna elements,the iirst element being located at the position N, the second elementbeing located at the position N2, and so on. Either approach is equallyvalid, and it makes no difference which is used in the end result in thearray.

To understand the rest of the derivation, it is well to give a generalexplanation of the technique which will be employed to create the array.Suppose, for example, that there are a given number of elements, sayfour, in an existing array. For any one of these four elements, anotherelement can be added, that is, a fth element, such that at any givenspace angle 0, the radiation from the fth element would be exactly outof phase, that is, out of phase, with the radiation from the rst of thefour elements. Likewise a second element, that is a sixth element, maybe located a certain distance from the second element of the rst four,so as to produce radiation which is exactly out of phase from theradiation produced by the second element of the original array of fourelements. In other words, if the position of the first four elements ofthe existing array were Vgiven as N1, N2, N3, N4 then to extend thearray by adding elements which produce zero radiation along a givendirection in space, that is, at a given value of 0, elements would beadded at distances determined as follows:

Thus, the position of the iifth element, that is, the position N5, isequal to the position of the rst element N, plus an added increment ofdistance S1. S1 is so chosen to result in radiations which arecompletely out of phase from the elements N1 and N5.

Likewise to provide a null in the radiation pattern, that is, zeroradiation intensity at some value of 0, it is necessary to place anadditional element the same incremental distance Sl from each of theoriginal elements in the array. Thus, in Equation 5:

Two more elements N7 and N8 are likewise placed in the same manner fromthe elements N3 and N4. This tech` nique of building up the array allowsa design wherein at certain points in space at certain values of thekangle 6, nulls'are produced in the radiation pattern. It remains howeverto provide a systematic manner of choosing the incremental distances,such as S1, which must be added to the existing members of the array toproduce the next set of members to increase the size of the array.

' .Referring `again either to Equation 1 or Equation 3, we can writeEquation 6:

Where ZN=ANeJ`NX,b, a general term of the series given iu Equation l orEquation 3.` For -any particular element -in an existing array, a-second element can be placed a particular distance from this {rstelement, such that the radiation is 180 out of phase. This is themeaning of Equation 6. Therefore, lfor the next element, that lis NCLH),the relationship given in Equation 7 prevails:

Thus, the neX-t element is given in terms depending upon the previouselement. In the quantity 2d sin 6 the quantity 2 occurs because anout-of-phase condition has been chosen.

:It is now necessary to provide a method of picking S1 (of Equation 4)in terms of some design parameter that will allow a relatively easymanipulation of the design lparameters with a minimum of calculation. Atthe same time, lit is desirableto pick the design parameter in such amanner that the .total etfect of all of the elements in the array whenit is considered -at any particular given value of 6 can be predicted. Inow choose anarbitrary design factorv K -in such ya manner. ConsiderEquation 11.

l Let -d sin 6 Equation 11 is true by deiinition, that is, K was chosenin thismannen From Equation 77 substituted back into Equation 10,Equation 12 Ifollows:

12) Then N X+D=NX+1T lEquation 12 states that to form a new elementposition CN(X+1), we add the Vquantity 1/ 2K to the previous elementposition NX. In other words, S1 equals 1/2K1. Equation 12 thus indicatesthe basic method for building up the array. Each time it is desired toextend the array by increasing the number of elements, the size of thearray will be doubled. lFor each existing element, one more element willbe added. Each of the added elements will be the distance 1/2K from itsown corresponding previous existing element.

-However, the quantity K can still be chosen in such a manner as tofacilitate design procedures and to make as' simple Vas possible thecomputations involved. For this reason, the following steps are taken.Eirst, 60 is dened as the physical space angle where the rst null of theantenna radiation pattern occurs. Now K1 is made equal to 1 at the dirstnull when 6 equals 60. By doing this,

the entire design of the array is normalized This will become moreapparent as the explanation proceeds. With K equal to 1 and 6 equal to60, then Equation 1l becomes Equation 13A.

)t 3 (l A) 1 d sin 6 l Equationl3A may n-ow be rewritten as Equation13B:

(13B) *sin 60 be obtained which show that K equals the ratio of sin 6divided -by sin 60:

(14A) t sin o (14B) 1p sin 6 d sin 6 A sin 6 (15) K" t sin o., a

sin 6 (16) *sin 60 In other words, K is the ratio of the sine of anyparticular value of the space angle 6 divided by the sine of theparticular space angle where the null of the first lobe occurs, that is,the main lobe of the antenna radiation pattern. Thus Equation 14B can berewritten by substituting K for sin 6 sin 60 as given by Equation 16:

1{/=(360)K The actual physical position of an element is equal to which,when the above value of yb is substituted, may be written:

sin 6 goNxilf-soNxK-wx sin 00 Now the actual physical coded spacings ofthe antenna array occur for sin 6:1. In other words at 6 equal to Thisis because at 6 equal to 90, the array is being examined directlybroadside, that is from a side elevational view. Thus Equation 18 givesthe physical coded space relationship.

360 Nsin 6 Equation 18 is given in units of wavelengths, that is,depending on .the particular value sin 60 which is chosen, an actualphysical array can be built. 60 -it should be remembered, is the anglein space where the rstnull occurs `for the antenna radiationV pattern.Equations 12, 17 and 18 provide the basis .for computing al1 of therequired quantities to buid the array with any arbitrary antennapattern.. To illustrate this, Equation 12 will be rewritten providing anactual index X which indicates how the particular values of K which havebeen chosen as arbitrary design factors are incorporated to build up thenext step of the array.

The basic procedure is this: To forman array of larger size, the numberof elements in the array must be doubled. This is because each existingelement of the previous array must have added to i-t, that is, placedsome distance from it which we have denoted by S, another element(whichv N ab will produce at some particular space angle 0, theradiation which is 180 out of phase with the radiation from thiscorresponding element of the previous array; Thus, the number ofelements in the -array must be a power of 2. An array built according tomy method therefore may consist of 2 elements, 4 elements, 8 elements,16` elements, 312 elements, 64 elements, and so on, but the number ofelements in the array will always be a power of 2. Likewise, for eachladditional power of 2. 'which is created when the size of the array isdoubled, -another arbitrary value for the design constant K may bechosen. =In other iwords, in an array which has 16` individual antennaelements, four values of K may be chosen arbitrarily. This is because 2to the 4th power is equal to 16, and the array is in a sense composed offour sets of elements which have been spaced in relationship to eachother to produce the result according -to my method.

Equations 19, 20 and 21, written below, symbolize this procedure. -InEquation 19, N1 is the iirst element of an existing array. T o increasethe size of the array, another element must be added. This is the term1N 2X+1) which is the iirst new element of the next set which is beingadded to the array. This term N 2X+1 represents lan element which mustbe spaced a distance 1/2K(X+1) from N1.

Stated in another way, Equation l19 shows how to locate the Ifirstadditional element when the array is being increased from a given size.Likewise, Equation 2O shows how to add the second new element of theincreased array. Equation 21 shows how to add the last element toincrease the array to its iinal size.

It should be noted that the subscripts yare powers of 2. X is simply anindex number which takes on integral values as the size of the arrayincreases. For an array which has two elements, X is equal to tor anarray that has four elements, X is equal to l; for an array that has 8elements, X is equal to 2; for an array that has 16 elements, X is equalto 3; :and so on. Every time the size of the array is doubled, that is,every time X increases one unit, this allows one additional value of Kto be arbitrarily chosen.

Where X=0, 1, 2, 3, etc.

A discussion of the dirst embodiment of our invention shown in FIGURE 2and some examples will make clear the procedure which can actually beperformed extremely rapidly with pencil .and paper in most cases.Referring to FIGURE 2, there is shown a linear array antenna composed of16 individual antenna elements numbered 1 through 16. This drawing is incorrect relative scale, that is, the correct relative spacing `of allthe elements is actu-ally given in FIGURE 2. A scale is provided innormalized electrical degrees running from 0 degrees, which is thereference value of the tirst element :1, up to 450. The location of thelast element 16 is at 435 electrical degrees. The values used for K tocreate this may are K1 equal to 1, K2 equal to 1%, K3 equal to 2 and K4equal to 4. Thus with an array of 16 elements, four values of K may bechosen arbitrarily. A translation device 17, which might be atransmitter or receiver, for example, is shown connected to the 16elements of the array by feed lines such as 18,\.18a,l18b,and so on.Equal amounts of power are supplied to each one of the 16 elements inthe antenna array; likewise the phase of the signal from the device 17is exactly the same for each of the 16 elements.

I shall now illustrate [with some examples how this array of FIGURE 2 isdesigned and will show the re- S sultin-g design'curves and theresulting radiation pattern in space. Next to each one of the elementsin the array, the identifying position notation such as N1, N2 etc. hasbeen written, so that the elements of the array can be identiiied. p |Itshould be noted that, although once the array is built, it makes nodifference what any particular element is called, in actually designingthe array, the numbering of the elements is important until the designhas been coma pleted. This is the reason that element N9, for example,is next to element =N1. Likewise the next element proceeding from leftto right -is N5 and then N3. 'I'he reason for this will be made clear bythe example. In actual physical constructions, since all of the elementsare the same, and since each is ifed with the `same amount of power atthe same phase, it makes no difference whether or not their notation ispreserved.

Code No. 1 is given below:

TABLEHI Code N0. 1 (shown in FIGURES No. 2 and No. 5)

16 element code with forced nulls `chosen at:

K1=\1, K2'=072, K3=2, K4=4 Actually the values listed are N1, .N2 etc.in this table are ipN1, gbNz. In other words, these are normalizedelectrical degrees and they do not yet represent physical distances,although they tare in the cor-rect relative proportion to each othertorepresent physical distances. I have used a shorter notation, such asN1, for the berivity of presentation. These values here are the same asthose shown in FIGURES 2 (and 5), as can be veriiied by the reader bysimple comparison. 11 lwill show by example how these values for thisiirst antenna array were obtained. First, for convenience, the iirstelement of the array N1, element l1 of FIGURE 2 is taken as having zerophase, that is, it is the reference element.Y All other elements will bepositioned in reference to this first element. Utilizing Equation 19, itis readily perceived where the next element is to be placed. The nextelement N2 will now provide an array of t-wo elements, this allowsexactly one value of Kto be chosen. K has been equal to 1, as indicatedbefore, .to provide a normalized design for convenience. Thus, fromEquation 19, it may be seen in Equation 22B, below, that the secondelement should be placed '180 out of phase with the rst element.

This is actually the basic technique of our invention: to place eachsuccessive element out of phase with its corresponding element from theprevious array. It should be noted that Equations 19 and 22A actuallygive the calculation of the positions in terms of the design parametersK. K, in the present instance equals l. Multiplying by 3,60 thenconverts into electrical degrees, sinceV there are 360 electricaldegrees for one wavelength of radiation. However, this is an arbitrarycalculation, and

if it is more convenient, the values may be left in the form shown inEquation 22A, namely 1/2, for example,

and the conversion need be done only atv the end. For

the time being, the readers visualization is greatly assisted byconverting into electrical degrees.

It should also be remembered that these electrical degrees to beconverted into the distances in space depend Obviously on the actualoperating frequency chosen for the antenna, since the wavelength changesphysically for different frequencies. However, the values given in theTable I also depend upon the particular design parameter 60 which canalso be chosen at will, as was indicated in Equation 16. This will alsobe made clear by the later discussion.

So far the array consists of two elements, namely, N1 and N2. It is nowdesired to increase the size of this array to four elements. To do this,consider Equation 19. The index X is now equal to l, in other words, asecond value K2 for the designed parameter K can be chosen. I havechosen K2 to equal 3/2. Thus from Equation 19, Equation 23 can bewritten:

The position of the third element of our array can be calculated. Itshould be remembered 'that each time a power of two is reached in thenumber of elements in the array, it is necessary to start over and beginadding elements starting from the irst element. In other words, thethird element is added to the rst element. The fifth element will beadded to the iirst element; likewise, the sixth element will be added tothe second element. This is because an array must consist of a number ofelements which is a power of two due to the design procedure which isutilized. Thus N3 is equal to 120, since K2 was chosen to be 3/2. .If K2had been chosen a different value, the position of element N3 would ofcourse be different.

The reader, by examining these equations, will readily perceive thatthey are simply special cases of Equations 19, 20 and 21 which definethe procedure for calculating the additional elements of the array.Thus, when the array reaches 4 elements, the fth element is calculatedusing the third value, K3, for the design parameter K. Likewise when thearray reaches 8 elements, to add the ninth element, the next arbitraryValue K., for the design value K is used. Here K3 is equal to 2 and K4is equal to 4. The only limitation on the choice of the design values Kis that they be numbers greater than one, this is to keep the designnormalized with K1 equal to 1. Other than that, K values may be chosenas integers, fractions, irrational numbers and so on.

This also provides an opportune place to point out the distinctadvantage of my design procedure. Suppose,

10 l for example, that the design of an array is proceeding and threevalues of K have been chosen, namely, K1 equal to l, K2 equal to 72, andK3 equal to 4. There are now eight elements in the array, it is desiredto add eight more elements and create a 16 element array. An inspectionof the plot for the eight elements reveals that it would be desirable toplace K4 equal to 2. This is readily done and creates no diiiicultywhatsoever. The 16 element array which results will be exactly the sameas that shown in Table I and it makes no difference in what order theparticular values of the Ks are chosen. The reader can readily verifythis for himself by calculating the elements using the same set of KValues but in a different order.

If the four values for K are chosen as l, 4 and 2 in any order, thenumbering of the elements will change in general. However, the relativespacing of the 16 elements will result in an array that is exactlyidentical to that shown in FIGURE 2 for any ordering of the design Ks.This can be veried by the reader himself in a few moments ofcalculation. n

Thus, at each step in the array, if the array is increased in size, thedcsignercan review the radiation pattern produced by the number ofelements used up to that point and he can pick the additional elementsand place them so as to provide an improvement in the radiation patternand never a degradation. This is a distinct advantage over the prior artdesign procedures. The addition of Imore elements to the array alwaysimproves the radiation pattern and canV never degrade it.

Likewise, there is another feature of great importance in my method ofbuilding antennas. array is to be made up of 32 elements. Using theprocedures outlined above, the first 16 elements of the array arecalculated and their locations noted. Now, one more value for K5 is tobe chosen. If the lirst four K values are not changed, the position ofthe first 16 elements of the array is in no Way affected by the additionof the next 16 elements. Thus the benelicial results obtained from therst part of the calculations will never be lost or degraded by theaddition of additional elements to the array. This is an unusual resultand obviously extremely advantageous. Those skilled in the art willappreciate the design and practical advantages of being able to addelements to an array with an assurance that the pattern already obtainedcan not be degraded, but only improved by the addition of more elements.An example will be given in connection with FIGURE 6 which shows a 32element array.

I have now explained my design procedure and the process for determiningthe coded spacings for my antenna elements. For future reference, I nowtabulate in algebraic form the general equations for calculating thepositions of the elements of an array up to and including 64 elements,that is, 2 tothe power 6, which allows the choice of six values for K,the arbitrary design factor. These equations are simply Equations 19, 20and 21 written out in algebraic form where each previous element is keptin the algebraic form involving the previous design values chosen for K.These 64 equations are unchanging and they present the general equationsfor calculating the position of any size array up to'64 elements, thatis an array of 2, 4, 8, 16, 32, or 64 elements. The actual positions, ofcourse, depend upon the values chosen for the design parameter K.Although the 64 equations presented here, numbered 33 through 96, appearto be in a rather awkward form, actually there is a practical advantagein presenting these calculations in this manner, rather than using thesimpler form of calculation indicated by equations, such as 19. Thiswill be pointed out in particular with reference to FIGURE l0 whichshows the left half of a 64 element array.

Returning now to FIGURES 2 and 3, FIGURE 3 shows the plot of theradiation intensity Et of the antenna of FIGURE 2, plotted versus thedesign factor K. It should be stressed that this plot is in K and notyet in Suppose that an` space angle which will illustrateanother'important advantage of our method and system. It can indeed beseen that nulls occur in this pattern, that is, zero values of radiationat values or" K equal to 1, 2, 11/2, and 4, as was indicated by ouroriginal choice of the values of K1, K2, K3, and K4. In general, one ofthe advantages of our design procedure is to form the design usingplots, such as FIGURE 3, where the abscissa is in units of K. Thisgreatly facilitates the design, and it results in a normalized designwhich can be adapted for a number of other conditions by certainphysical spacing when it is actually constructed.

It may be noted in FIGURE 3, that the central lobe is extremely sharpand contains a large percentage of the total energy emitted from thearray. The total integrated area under the graph represents the totalenergy emitted by the 16 elements of the antenna of FIGURE 2. Once aparticular code has been picked, utilizing given values for K, theradiation pattern can be plotted in terms of K by utilizing Equation 3which is now rewritten as Equations 98A, 98B and 98C.

all N 2 all N 2 (98C) EF cos Nif) +(Z sin Na) all N where E signifiesthe summation for all n terms present in the code;

all N 2 all N 2 In Equation 99, E, is given as a function of K. To formthe plot, a value for K is picked, then Equation 99 is calculated andyields a value for Et. Then K is increased a convenient increment andEquation 99 is recalculated forthe new value of K, and so on, for theparticular array which has the fixed relative spacing depending upon thedesign parameters of the constants K1, K2, K3 etc. which have beenchosen. This plot of E, versus K is shown for the array of FIGURE 2 inthe illustration of FIGURE 3. To evaluate Equation 99, only values of Kwithin a region up to a value of KMAX equal to l/sin 00 need be chosen.

In FIGURE 4, the radiation pattern of the antenna of FIGURE 2 has beenshown plotted against 0, the physical space angle. To form this plot sin00 was chosen as equal to 1/6. In other words, @o equals 9.6"' Thisrepresents the width of the first lobe, or the main lobe, that is, thepoint along the 0 axis where the first null occurs, as shown on thediagram. It should be noted that, since sin 0 is equal to K sin 0G,there is no linear correspondence between the values of K and the valuesof 0, even for a fixed value of 00. This is because 0 and K sin 00 arerelated by the sum of 0 and not one to one. This causes no computationsdiiculties, however. It should also be noted that a variation of 0through 90 physical degrees is sufiicient to specify the fullperformance of the antenna pattern as was mentioned previously. Anexamination of FIGURE 3 shows that nulls occur in fact at values of Kequal to l, W2, 2 and 4. This is as expected, since these Vpoints werepicked to be forced nulls according to the principle of my method ofdesigning array antennas.

FIGURE 5 shows a second embodiment of my invention which illustrates theextreme practical utility of my antenna arrays and the method ofbuilding them. The gure shows a similar codedV array, as in FIGURE 2,extended into a two dimensional array. It is a plan view of the array,and the individual antenna elements have been simply shown as dots forconvenience. The correct relative proportions are shown and the scale isthe same as in FIGURE 2. FIGURE 5 was created from FIGURE 2 with nofurther computation. The array'antenna of FIG- URE 5 provides an antennaradiation pattern in three dimensions in space, that is, not only doesit provide the radiation pattern shown in FIGURE 4 in the plane of theaxis of the array of FIGURE 2, but it provides this same pattern in aplane perpendicular to the plane containing the axis of the 16 elementsof FIGURE 2.

An inspection of FIGURE 5 will quickly reveal its pattern. The first rowacross, consisting of 16 individual antenna elements, which might bedipoles, dicones, horns etc. or whatever is convenient, are numbered 1through 16 and are numbered corresponding to FIGURE 2. It will be seenthat the second row of antenna elements, relative to each other, isspaced exactly the same way as the first row. Thus, element 19 which isthe first element of the second row is 45 electrical degrees fromelement 20 which is the second element of the second row. Likewiseelement 21 is exactly 45 electrical degrees from element 20, just aselement 5 is 45 electrical degrees from element 9 in the rst row.However, the entire second row, such as elements, 19, 2t), 21, 22 and soon, is spaced the same distance from the first row as element 9 isspaced from element 1, namely 45 electrical degrees. The rst element ofthe third row, element 23, is the same distance from element 19 aselement 5 is from element 9 in the first row. Likewise element 24, whichis the iirst element of the fourth row, is the same distance fromelement 2'3, as element 3 is from element 5 in the iirst row. Alsoelement 25, which is the second element in the fourth row, is the samedistance from element 24 as element 9 is from element 1 in the iirstrow. The pattern is extended in this manner, as shown, as it will beseen that along any line of elements, either horizontal or vertical, therelative spacing is exactly the same as that shown in FIGURE 2 or therst row in the array of FIGURE 5.

Thus, the spacing of the array along a single axis, such as shown inFIGURE 2, in actuality also completely spaces the location of all theelements of a square array consisting of nXn elements, where n is thenumber of elements in the single axis array. The total number ofelements, shown in FIGURE 5 is 16 squared, that is, 256 elements. Y Y YY Y This property of my antenna arrays, that they may be reproduced intwo dimensions to create a three dimensional antenna radiation patternis obviously of great practical utility and is true of all antennaarrays created according to my method, and Vaccordingly the embodimentsof FIGURES 6 and 10 also may be extended in two directions in the samemanner.

For convenience of illustration, I have only shown the embodimentsofFIGURES 6, 8 and 10 along a single axis. But the embodiments ofFIGURES 6, 8 and 10 may likewise be extended into a two dimensionalsquare array in the same manner, and I claim such two dimensional arraysas part of the novelty of my invention.

Just as in FIGURE 2, the antenna elements of FIG-Y URE 5 should besupplied with equal amounts of power from a source providing the samephase of signal to each of the elements. ForV convenience, thetranslation device, such as 17, has not been shown in FIGURE 5b.

For such a relatively large array as is shown in FIG- URE 5, it may bemore convenient if a separate power transmitting stage is connected toeach of the elements and supplied with a low power level signal from thesame local oscillator, for example, as indicated schematically in FIGURE5b.

FIGURE b merely illustrates what has been mentioned before, that each ofthe individual antenna elements of my array, such as N1 or N9 or N7 canbe individually supplied with a high power transmitting stage forsending purposes for use of the array as a transmitting antenna and withan individual amplier for using the array for receiving purposes. Eachof the power transmitting stages, such as 26, 27, 28, and so on, can beexactly identical in design; each transmitting stage, 'such as 26,supplies the same amount of power to its individual antenna element asall the other transmitting stages. Thus, these transmitting stages mightconsist of an individual travelling wave tube, for example, With anassociated power supply and frequency control circuits.

As a result, these individual transmitting stages can each be made tooperate at its peak point of design eiciency, and there is no necessityfor wasting RF power in power dividing networks for providing unequalamounts of power to the individual elements.

Likewise for receiving purposes, if desired, each individual antennaelement such as N1 may be provided with a high gain, low noiseamplifier, such as 29, 30, and 31. Each of these amplifiers is exactlythe same as the others and provides an equal amount of gain. This alsoallows the output from each individual antenna element to be amplifiedimmediately before passing to the connecting lead networks such as 32,where it is connected to the input o'f the rest of the receiverequipment.

Obviously, if desired, the ampliier, such as 29, and transmitting stage,such as 26, may be omitted and the array of FIGURE 5 may be suppliedwith one single transmitting or receiving apparatus, as indicated inFIG- URE 2, by the use of multiple feed lines, such as shown as 18, 18a,18b, and so on. Whichever is most convenient for a particularapplication will be used. The individual amplifiers and transmittingstages, such as 26 and 29, can be physically disposed extremely close tothe individual antenna elements in the arrays built according to mymethod.

Refer now to FIGURE 6, which shows the correct relative spacing for anarray antenna built according to my method and composed of 32 individualantenna elements, spaced according to Table II.

TABLE II Code N0. 2 (shown in FIGURE 6) 32 element code with forcednulls chosen at:

`when building up Code No. 2 with a 32 element array.

The fth value K5 has been chosen as 5A.. The quan- 14l tity 1/2K5 isthus equal to 144 electrical degrees and the 16 new values for Code No.2 may be formed by adding 144 increments to the appropriate terms inlCode No. 1, as has been outlined in the previous development,particularly Equations 19, 20 and 21. Alternatively, the code elementpositions may be calculated, using Equations 33 through 55.

FIGURE 7a shows the rst part of the plot of the radiation intensity Etversus the design parameter K of the antenna of FIGURE 6. FIGURE 7bshowsy the same antenna array plotted out to values of KMAX equal to 30.FIGURE 7c will be seen to be an expanded version near the origin ofFIGURE 7b. FIGURE 7c also shows Code No. l having 16 elements and CodeNo. 2 with 32 elements plotted on the same scale for comparison.

It is important to point out at this time a further additional advantageof my method of building antennas. My design, as so far explained,permits a given number of values for the design parameter K to be chosenin an arbitrary manner, depending upon the number of elements in thearray. At these chosen values of K, such as K1, K2, K3 and so on, therewill be zero radiation in space at the corresponding space angle 0. Itcan now be stated that in addition there will be a null at every oddintegral multiple of the chosen values of K. Thus, if K1 is equal to 1,nulls will also be produced at values of K equal to 3, 5, 7 and so on.Likewise if K2 is chosen to be 5%?, there will be nulls produced at 4.5,7.5 and so on. Inspection of FIGURE 3 will clearly reveal these nullswhich have been circled lightly with dotted lines. This property of thepresence of nulls at all odd integral values for all the chosen valuesof K, in eiect allows double duty to be performed by the chosen valuesof K.

The values of K which are chosen depend on what type of radiationpattern is to be produced. In the examples given so far, the antennaradiation pattern has been of the type where it is desired to produce acentral lobe which has as high a maximum value relative to the sidelobes as possible. In addition the central lobe is intended to be asnarrow in angular degrees as possible. Further, the amplitude of thesecond and third side lobes is intended to be as low as possible, and,in addition, the occurrence of the second and third side lobes isintended to be pushed out as far in angular degrees as possible.

By comparing FIGURE 3 with FIGURE 7a, and by referring to FIGURE 7c,which is the combined plots of FIGURE 7a and FIGURE 3, the effect ofadding 16 additional elements in Code No. 2, to produce an array having32 elements can be appreciated. Thus, in FIG- URE 3, the peak of the rstside lobe occurred approximately at a value of K equal to 1.25, as canbe seen. Note, that in Code No. 2, FIGURE 7a, K5 was chosen to be equalto 1.25, that is, 5%. Thus, a new additional null was chosen to be rightin the middle of the first side lobe. In effect this squashed down therst side lobe. Examining FIGURE 7a near the region 00, it can be seenthat the rst side lobe, and in fact the second side lobe, have virtuallydisappeared, their amplitude being so small relative to the main lobethat it is dicult to show it.

It can also be seen in FIGURE 7a that an additional null has beenproduced at K equal to 3.75. This is 3 times 1.25. This additional nullat 3.75 had the effect of considerably reducing the magnitude of thefourth side lobe, as can be seen on FIGURE 7c, by comparison of the twocurves.

Also note in FIGURE 7c that the sharpness of the main lobe of the 32element array of Code No. 2 has also been improved.

It can be appreciated that Code No. 2 has greatly improved the radiationpattern by the addition of more elements in a controlled manner chosenat the designers will. Likewise, I wish to point out that antennaradiation patterns of virtually arbitrary shape can be produced by mymethod. For example, it may be desired `to concentrate a side lobe powerin some angular region in space that is in a certain range of values for0. To do this, it is only necessary -to place nulls outside of thisregion before the beginning of the region and after the end of theregion, the resulting power then must appear in the region where nullswere not chosen. In other words, by being able to choose points in spacethat have nulls, in effect the maximums of the radiation pattern mustexist. in other regions in space, since the total energy supplied to theantenna cannot change, and in effect the designer can locate maximums ofradiation by specifying where the minimums occur.

I now wish to point out another feature of my invention vwhich is quiteconvenient from the viewpoint of physical construction and also as acheck of the accuracy of computations. Examination of FIGURE 2 will showthat it is precisely symmetrical about the point located at 217.5". Thisis exactly the middle of the array, the furthermost element N16 beinglocated at 435 electrical degrees. This physical symmetry is alwaysproduced, although the equations given would not indicate it bythemselves.

This physical symmetry of the codes also indicates that the method ofderivation rests on solid physical It will be seen that two. of the Kvalues are the same, namely, l and However, in Code No. 3,'FIGURES 8 and9, K2 has been chosen as 5A and K4 has been chosen at 11/6. It can beseen from inspection of FIGURE 9 that the first side lobes haveimmediately been tremendously reduced by this expedient. In Table IIIpresenting Code No. 3, I have simply presented the relative spacinglocation of the elements in order of increasing value, their computationis exactly the same as that given in Equations 19, 2() and 2l, but theelement numbering according to the equations has not been preserved. Todistinguish that the elements are arranged simply in order of increasingphysical location, I have labeled the locations as P1, P2 and so on, inTable III.

FIGURE 10 shows a preferred embodiment of my invention. Table IV givesCode No. 4 which gives the relative physical spacing which was plottedto scale in FIGURE 10.

TABLE IV foundations, because, as noted, the space angle 0 has only beenconsidered over 90, since the array was stated to be one which willproduce symmetrical radiation patterns to begin with. Likewise, incomputation, when an array has been completed, if its is plotted out toscale, as shown in FIGURES 3, 5, 8 and 10, and, in fact, the antennapattern is not symmetrical, then a mistake has been made in thecalculations.

This may suggest to the reader a method of saving computations bycomputing only the equations necessary for one half of the array andthen simply duplicating the array about the axis of symmetry.Unfortunately, however,` it is not usually readily apparent whichparticular elements will lie on one particular side of the array; thisis because physically, as the elements are calculated, they areinterleaved with the previous existing elements of the previous arrayand as a result a higher numbered element does not necessarily liefarther from the rst element than a corresponding lower numberedelement. Thus, for example, element N9 is actually much nearer 45 toelement N1 than element N3 is, and so on.

However, once the array has been calculated completely, the relativespacing may be arranged arbitrarily in increasing order of distance fromthe rst element N1. Then the rst half of the set of values may beplotted out, and this will produce the iirst half of the antenna. Thesecond half of the antenna will be exactly similar to this and in factbe the mirror image.

FIGURE 8 shows another embodiment of my invention also utilizing'a 16element array. However, the values of K chosen for this array aredifferent from that shown in Code No. 1. The antenna utilizing Table IIIwith Code No. 3 has been drawn in FIGURE 8 to scale and the resultingradiation pattern as a function of K is shown in FIGURE 9.

The antenna radiation pattern as a function of K has been given inFIGURE llfor the antenna shown in FIG. URE 10 and Table IV, and FIGURE12 shows the antenna radiation atterri as a function of the physicalspace angle TABLE III 0 for a vague of sin 00 equal to arc sin M5,namely 9.6.

Code No 3 An inspection of FIGURE i2 wiii immediateiy make 16 elementCode with forced nuns chosen at: apparent to those that are skilled inthe antenna art that K1=1 [(3:572 65 this antenna radiation pattern isquite unusual. It can be Kzzf/g Kzll5 seen that the main, that is thecentral lobe, is quite sharp in angular degrees and in fact the width ofthe main beam P1=0 P9=278-18 at the one half power point is only 5angular degrees. P2=98-18 P10=3O0 In addition, the rstvtwo side lobesare so low in magni- P3=120 P11=324 70 tude they can hardly be plottedto the same .scale as the P4=144 P12=362-18 main lobe. Further, the rstside lobe which has an l P5: 180 P13=398,18 appreciable amplitude, thethird side lobe, haso been physi- P6=218-18 P14=422-18 cally moved outto the region beyond 20 Such an P7=242-18 PM2/444 antenna has anextremely sharp beam which would b e ex- P3=264 75 cellent, for example,for tracking radar service or highly

1. A LINEAR ARRAY ANTENNA COMPRISING N INDIVIDUAL ELEMENTS, WHERE N ISAN INTEGRAL POWER OF THE BASE 2, AND MEANS FOR SUPPLYING THE SAME AMOUNTOF POWER IN THE SAME SIGNAL PHASE TO EACH OF SAID N INDIVIDUAL ELEMENTS,THE SPACING BETWEEN SAID INDIVIDUAL ELEMENTS BEING DETERMINED BY THEVALUES SELECTED FOR K(X+1), WHERE K IS A DESIGN PARAMETER AND IS THERATIO OF THE SINE OF ANY PARTICULAR VALUE OF THE SPACE ANGLE 0, FORMEDBY THE AXIS OF THE